B. Number · Number Sense: Numbers to 1000 and Fractions

Building and Breaking Apart Numbers in Many Ways

B1.1 B1.5 A1About 35 minutes

Learning goals

I can trade 1 hundred for 10 tens, and 1 ten for 10 ones.
I can show the same number in more than one way, like 243 = 2 hundreds 4 tens 3 ones = 24 tens 3 ones.
I can find a number made from unusual groups, like 3 hundreds 12 tens 6 ones.
I can break a number into parts that add up to it.

Words to know

compose
To put parts together to make a number.

decompose
To break a number into parts.

trade (regroup)
To swap 10 of one place for 1 of the next place, or the other way around. 10 tens can be traded for 1 hundred.

loonie
A Canadian one-dollar coin.

At the bank

Uncle Raj wants to take $243 out of the bank. The bank has $100 bills, $10 bills and $1 coins (loonies).

The usual way is 2 hundreds, 4 tens and 3 ones. But what if the bank has only one $100 bill left? Can it still give him $243? Yes! It can give more $10 bills instead. Numbers can be built in many ways.

Trading

You already know:

1 hundred = 10 tens
1 ten = 10 ones

So we can trade a hundred for 10 tens, or a ten for 10 ones. The amount stays the same. Only the way we show it changes.

Hundreds	Tens	Ones	Check
2	4	3	200 + 40 + 3 = 243
1	14	3	100 + 140 + 3 = 243
0	24	3	240 + 3 = 243
2	3	13	200 + 30 + 13 = 243

Four ways to make 243. Each row adds up to 243.

1 hundred, 14 tens and 3 ones. This is another way to show 243.

Breaking numbers into parts

We can also split a number into any parts that add up to it, not just place-value parts:

243 = 200 + 43
243 = 240 + 3
243 = 100 + 143
243 = 220 + 23

Breaking numbers apart like this will help you add and subtract in your head.

Worked example

Example 1: Unusual groups

What number is 3 hundreds, 12 tens and 6 ones?

3 hundreds = 300.
12 tens = 120.
6 ones = 6.
300 + 120 + 6 = 426.
Check by trading: 12 tens = 1 hundred and 2 tens. Now we have 4 hundreds, 2 tens, 6 ones: 426.

The number is 426.

Worked example

Example 2: How many tens?

How many tens are in 580?

Each hundred is 10 tens. 5 hundreds = 50 tens.
580 also has 8 more tens.
50 tens + 8 tens = 58 tens.

There are 58 tens in 580. (Look: 58 is the first two digits!)

Worked example

Example 3: Show 405 in three ways

Show 405 using hundreds, tens and ones in three different ways.

The usual way: 4 hundreds, 0 tens, 5 ones.
Trade 1 hundred for 10 tens: 3 hundreds, 10 tens, 5 ones.
Trade all the hundreds for tens: 40 tens, 5 ones.

4 H 0 T 5 O, 3 H 10 T 5 O, and 40 T 5 O. Each one is 405.

Worked example

Example 4: Packing crayons

A factory has 352 crayons. Big boxes hold 100 and small packs hold 10. The factory is out of big boxes. How can it pack the crayons?

With no 100 boxes, use packs of 10.
352 = 35 tens and 2 ones, because 3 hundreds = 30 tens, plus 5 tens = 35 tens.
So it fills 35 packs of 10 and has 2 crayons left over.

35 packs of 10 and 2 loose crayons. 350 + 2 = 352.

The factory's 352 crayons packed with no big boxes: 35 packs of 10 and 2 loose crayons.

Check it

No matter how you show a number, you can always check it: find the value of each group and add. If you get the same total, it is the same number.

Just for fun

What did the hundred flat say after it was traded for 10 rods?

"I'm not broken. I'm just decomposed!"

Hands-on activity

Bank teller game

You need: paper, a pencil and scissors (ask a grown-up for help with scissors). Or use real or play money if you have it.

Make play money: cut paper into rectangles. Write $100 on 5 of them and $10 on 20 of them. Cut small circles and write $1 on 20 of them.
A grown-up is the customer and asks for an amount, like $243.
You are the bank teller. Count out the money the usual way.
Now the customer says, "Please give me more $10 bills." Make the same amount a different way.
Try a third way. Write each way in a chart: hundreds, tens, ones.
Switch jobs!

Math talk: "How do I know both ways are the same amount?"

Your turn

What number is 2 hundreds, 13 tens and 4 ones?

What number is 5 hundreds, 0 tens and 17 ones?

How many tens are in 300?

Fill in the blank: 461 = 3 hundreds, ___ tens, 1 one

720 = ___ tens

Show 156 in two different ways.

600 + ___ = 675

More practice: warm-up

1 hundred = ___ tens

1 ten = ___ ones

What number is 20 tens?

What number is 4 hundreds and 5 tens?

300 + 50 + 8 = ?

More practice: on your own

What number is 6 hundreds, 14 tens and 2 ones?

Raj has 31 tens and 9 ones. What number is that?

Fill in the blanks: 845 = 800 + ___ = 840 + ___ = 745 + ___

A store has 3 boxes of 100 pencils, 22 packs of 10 pencils and 6 loose pencils. How many pencils?

Is 4 hundreds, 9 tens and 15 ones the same as 505? Explain.

Challenge

Show 230 in every way you can using only hundreds and tens (no ones).

How many tens make 1000? Explain.

Lena has 4 $100 bills, 7 $10 bills and 12 loonies. A bike costs $485. Does she have enough money?

Check your understanding

Quiz

1 hundred is the same as:

What number is 3 hundreds, 15 tens and 2 ones?

How many tens are in 640?

Which one is NOT the same as 270?

563 = 500 + ___

What number is 1 hundred, 12 tens and 5 ones?

For parents and teachers

Big idea: A number can be composed and decomposed in many ways. Knowing that 1 hundred = 10 tens is the key to regrouping in addition and subtraction later this year. The curriculum (B1.1 and B1.5) asks Grade 3 students to compose and decompose numbers to 1000 and represent them in a variety of ways.

Common difficulties:

Writing digits side by side: 3 hundreds, 12 tens, 6 ones written as 3126. Have your child find the value of each group and add.
Not seeing hidden tens: children often don't realize 580 has 58 tens, not 8.
Thinking the "usual" way is the only right way. Praise different correct ways.

How to help: Use money. Ask: "Can you make $150 with only $10 bills? How many?" When paying, talk about trading a $10 bill for ten loonies. Encourage your child to check by adding values (a Strand A habit: checking that an answer makes sense).